Systems of Linear Equations

How to solve for 3 variables with 3 equations (variables x,y,z)

1) Get two of the equations and make the value of x (y or z) the same 2) Subtract these two equations, giving you an equation with just y and z (x and z or x and y) 3) Repeat these first two steps, but switch one of the original equations with the remaining original one, leaving you with another equation with just y and z (x and z or z and y) 4) With these two new equations, make the values of y (x or z) equal and subtract one from the other 5) This new equation will have just a z (x or y) value and you can solve algebraically to find the value of z (x or y) 6) Plug in this value of z (x or y) into one of the equations with only z and y (x and y or x and z) and find the value of y algebraically 7) Plug the values of y and z (x and y or x and z) into one of the original equations and algebraically find the value of x (y or z)

With 2 equations (ex. 2x+y=3, y-x=0) what is the simplest algebraic method to find x and y?

1) Multiply the equations so that the y (or x) values are equal 2) Subtract either equation by the other 3) With this value solve for x (or y) 4) Plug x (or y) into either of the original equations to solve y (or x)

How to solve for a 2 equations with 2 variables by using graphing.

1) Transform both equations so that "y" is on one side and everything else is on the other 2) Go to the "y=" section of a graphing calculator and type in the formula 3) Graph both of these equations 4) Find the intersection to give you the value of x 5) Plug in the value of x to find the value of y

If there are 2 different equations with two variables each, both with the same to values (ex. 3y-2x=5, y+6x=9), how many values of x and y are there?

There is only 1 value of x and 1 value of y

Solve 2x+y-2z=-10 -3x+5y+3z=28 x-2y+z=-2

Transform 2x+y-2z=-10 -(2x-4y+2z=-4) 5y-4z=-6 -3x+5y+3z=28 -(-3x+6y-3z=6) -1y+6z=22 5y-4z=-6 -(5y-30z=110) 26z=104 *z=4* -1y+24=22 -1y=-2 *y=2* 2x+2-8=-10 2x-6=-10 2x=-4 *x=-2*

Solve x/y=16 x+4y=40

Transform to x=16y x-16y=0 -(x+4y=40) -20y=-40 *y=2* x/2=16 *x=32*

Solve by graphing 3x+y^(1/2)=5 5x-8y=-27

Transform to y=(5-3x)^2 y=5/8x+27/8 By graphing, at intersection of lines: *x=1* y=5/8(1)+27/8 y=32/8 *y=4*

Solve 3x-y=2 -4x+3y=14 by graphing

Transform to y=-2+3x y=4/3x+14/3 By graphing, at intersection of lines: *x=4* -16+3y=14 3y=30 *y=10*

How to solve a 2 variable equation using Cramer's Rule: 3x+4y=2 5x-7y=17

Transform to |2 ⋅⋅⋅4| ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ |17 -7| ⋅⋅-14-68 ⋅⋅⋅-82⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ x= ------ = ------- = ---- = *2=x* |3 ⋅⋅4|⋅⋅ -21-20 ⋅⋅⋅-41⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ |5⋅ -7|⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ |3⋅ 2|⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ |5 17| ⋅⋅⋅51-10⋅⋅⋅ 41⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ y= ------ = ----- = --- = *-1=y* -41 ⋅⋅⋅⋅⋅⋅⋅-41 ⋅⋅⋅⋅-41⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

Solve 2x+5y=17 6x+3y=15

Transform to 6x+15y=51 -(6x+3y=15) 12y=36 *y=3* 6x+45=51 6x=6 *x=1*